Burgers Riemann Entropy Shock and Fan

The inviscid scalar conservation law u_t + (u²/2)_x = 0, known as the inviscid Burgers' equation, provides a classic model for nonlinear wave propagation, shock formation, and rarefaction. A Riemann problem consists of a piecewise constant initial condition with a left state u_L and a right state u_R. Because the flux function f(u) = u²/2 is strictly convex, characteristics will either collide or spread out, leading to two distinct types of solutions.

worked-example

Let us calculate the analytical solution for a Riemann problem where characteristics collide, forming a shock wave.

Assume the initial left state is u_L = 2 and the right state is u_R = 0.
Because u_L > u_R, the characteristic lines x = u_Lt and x = u_Rt will intersect. To maintain a single-valued solution, a shock must form.
The shock speed s is determined by the Rankine-Hugoniot jump condition: s = (f(u_L) - f(u_R)) / (u_L - u_R). Since f(u) = u²/2, we have s = (2 - 0) / (2 - 0) = 1.
This shock satisfies the Lax entropy condition: u_R < s < u_L (0 < 1 < 2).
At time t = 1, the shock is located at x = s*t = 1. The spatial profile u(x,1) is 2 for x < 1, and 0 for x ≥ 1 (using our right-at-zero convention). A probe placed at x = 1 will therefore read 0.

experiment

We can observe both shock and rarefaction behaviors using the Burgers Riemann Entropy Shock and Fan interactive module.

Set the Left state to -1 and Right state to 2. Here u_L < u_R, so characteristics spread apart.
Set Time to 1. The characteristics form a rarefaction fan filling the space between x = -t (speed -1) and x = 2t (speed 2).
Move the Probe position to 0. Inside the fan, the solution is u = x/t, so the probe reads 0.
Change Time to 0. Notice the right-at-zero convention taking effect: the initial step is evaluated, so the probe at x=0 reads 2 instead of an undefined or averaged value.
Change both states to 1. This forms a constant state with no shocks or fans. The probe reads 1 everywhere.

limitations

This module considers only the inviscid Burgers' equation (zero viscosity). Real physical systems, like fluid dynamics or traffic flow, often involve viscous or diffusive terms that smooth out shocks over a finite thickness. The module also restricts the domain and values to a finite grid for educational visualization. Furthermore, while this model accurately captures the entropy condition for convex fluxes, non-convex flux functions would require a more complex convex hull construction (Oleinik's condition) which is not covered here.

references

For more information on the Burgers' equation and Riemann problems, see:

LeVeque, R. J. (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press. Chapter on Riemann problems and entropy conditions.
Clawpack Documentation on Burgers' Equation: https://www.clawpack.org/riemann_book/html/Burgers.html

Burgers Riemann Entropy Shock and Fan